Question

1. a truck drove 1200 miles to colorado city in 2 days. he drove 320 more miles on the first day than he did on the second day. how many miles did he travel on the 2nd day?
2. four of the five dogs in a pen weigh 35, 40, 12, and 68 lbs. if the average weight is 36 lbs, what is the weight of the fifth dog?
3. the length of a rectangle is 6 inches longer than its width. if the perimeter is 60 inches, find the length and width.
4. find three consecutive even integers with a sum of -318.
5. find four consecutive integers with a sum of 138.

Explanation:

Step1: Set up an equation for the truck's travel

Let the miles driven on the first day be \(x\). Then the miles driven on the second day is \(x + 320\). The total miles driven in 2 days is \(x+(x + 320)=1200\).
Simplify the left - hand side: \(2x+320 = 1200\).

Step2: Solve for \(x\) (first - day miles)

Subtract 320 from both sides: \(2x=1200 - 320\), so \(2x = 880\).
Divide both sides by 2: \(x=\frac{880}{2}=440\).

Step3: Find the second - day miles

Since the second - day miles is \(x + 320\), substitute \(x = 440\) into it. So the second - day miles is \(440+320=760\).

Step1: Set up an equation for the dogs' weights

The formula for the average \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here, \(n = 5\), \(\bar{x}=36\), and the known weights are \(x_1 = 35\), \(x_2 = 40\), \(x_3 = 12\), \(x_4 = 68\), and let the fifth weight be \(x\).
We have \(36=\frac{35 + 40+12+68+x}{5}\).
Multiply both sides by 5: \(36\times5=35 + 40+12+68+x\).
Calculate the left - hand side: \(180=35 + 40+12+68+x\).
Calculate the sum of the known weights: \(35 + 40+12+68=155\).
So \(180=155+x\).
Subtract 155 from both sides: \(x=180 - 155 = 25\).

Step1: Set up equations for the rectangle

Let the width of the rectangle be \(w\). Then the length \(l=w + 6\).
The formula for the perimeter of a rectangle \(P = 2(l + w)\). Given \(P = 60\), substitute \(l\) into the perimeter formula: \(60=2((w + 6)+w)\).
Simplify the inside of the parentheses: \(60=2(2w + 6)\).
Expand: \(60=4w+12\).
Subtract 12 from both sides: \(4w=60 - 12=48\).
Divide both sides by 4: \(w = 12\).
Then \(l=w + 6=12 + 6=18\).

Step1: Set up an equation for the consecutive even integers

Let the first even integer be \(n\). Then the second is \(n + 2\) and the third is \(n+4\).
The sum \(n+(n + 2)+(n + 4)=-318\).
Simplify the left - hand side: \(3n+6=-318\).
Subtract 6 from both sides: \(3n=-318 - 6=-324\).
Divide both sides by 3: \(n=-108\).
The second integer is \(n + 2=-108+2=-106\), and the third is \(n + 4=-108+4=-104\).

Step1: Set up an equation for the consecutive integers

Let the first integer be \(m\). Then the second is \(m + 1\), the third is \(m+2\), and the fourth is \(m + 3\).
The sum \(m+(m + 1)+(m + 2)+(m + 3)=138\).
Simplify the left - hand side: \(4m+6=138\).
Subtract 6 from both sides: \(4m=138 - 6=132\).
Divide both sides by 4: \(m = 33\).
The second integer is \(m + 1=34\), the third is \(m + 2=35\), and the fourth is \(m + 3=36\).

Answer:

1. The truck traveled 760 miles on the second day.
2. The weight of the fifth dog is 25 lbs.
3. The width of the rectangle is 12 inches and the length is 18 inches.
4. The three consecutive even integers are - 108, - 106, - 104.
5. The four consecutive integers are 33, 34, 35, 36.